Supreme Court justices appeared deeply conflicted Tuesday as they considered the fate of Baby Veronica, whose adoptive parents were ordered to return her to the Native American father who once gave up rights to his biological daughter but now has embraced parenthood.

The hour-long oral arguments featured an emotional appeal from the attorney for Matt and Melanie Capobianco, the adoptive couple, and a lament from Justice Anthony M. Kennedy about the difficulty the case presented.

“These considerations are why domestic relations pose the hardest problems for judges,” he said. “If we could appoint King Solomon, who was the first domestic relations judge, as special master, we could do it. But we can’t do it.”

The case turns on the Indian Child Welfare Act (ICWA), which Congress passed in 1978 to help preserve Native American families by erecting high hurdles for ending the parental rights of Indian parents and to discourage adoptions outside tribes.

Washington lawyer Lisa S. Blatt, representing the Capobiancos, told the justices that the intent of the law had been perverted in the case of Veronica and that affirming the decision to return the child to her father would set a dangerous precedent.

“You’re basically banning the interracial adoption of abandoned Indian children,” she told the court.

Moreover, she said, the lower court’s interpretation of the law gave an unwed Indian father veto power over a non-Indian mother’s decision to place her child for adoption.

“Your decision is going to apply to the next case . . . where a tribal member impregnates someone who’s African American or Jewish or Asian Indian, and in that view, even though the father is a completely absentee father, you are rendering these women second-class citizens” in deciding who raises their child, Blatt said.

Charles A. Rothfeld, representing Dusten Brown, the girl’s biological father, said it was simply false that a family court in South Carolina and the state Supreme Court did not decide the case based on the best interests of Veronica.

“The state courts found that ICWA should be applied to allow a natural father to raise his child,” Rothfeld said. “Those courts found that it’s in the best interests of the child . . . because that parent was a fit, was a loving, was a devoted parent.”

Brown and Christina Maldonado became engaged in December 2009 and learned a month later that she was pregnant. But she called things off after Brown pressured her to get married right away, and he later texted, in response to her question, that he would rather give up parental rights than pay child support.

Maldonado decided to place the baby for adoption. The Capobiancos met the mother through an adoption agency, supported her in her last months of pregnancy and were in the delivery room when she gave birth. They named Veronica and brought her to their home in Charleston, S.C.

When Brown later learned that the child had been placed for adoption, he immediately tried to stop the process and, as a member of the Cherokee Nation, invoked the ICWA.

The Capobiancos were ordered to turn over Veronica when she was 27 months old. She has lived with Brown and his new wife, Robin, in Nowata, Okla., for 15 months.

The couples sat in different sections of the ornate courtroom Tuesday as the justices applied a legal microscope to the provisions of the ICWA in search of an answer to Veronica’s fate.

Justices Antonin Scalia and Sonia Sotomayor had the sharpest questions for Blatt.

Scalia said it would be “extraordinary” for courts to consider whether it was in the child’s best interests to terminate Brown’s parental rights and allow the adoption.

Sotomayor told Blatt that if “the father’s fit, why do you think that the federal statute requires that [the child] be given to a stranger?”

Blatt responded that “the only stranger in this case was the birth father, who expressly repudiated all parental rights and had no custodial rights.”

That seemed to strike a chord with Chief Justice John G. Roberts Jr., who was skeptical of Rothfeld’s description of Brown as “excited” by news of the pregnancy.

“There is no doubt he paid nothing during the pregnancy and nothing at the time of the birth, right, to support the child or the mother?” asked Roberts, who has two adopted children.

Before Rothfeld could answer completely, Roberts interjected: “So he was excited by it; he just didn’t want to take any responsibility.”

Roberts and other justices wondered whether any amount of Indian blood was enough to invoke the ICWA. Blatt has calculated that Veronica is “3/256ths Cherokee,” which Justice Stephen G. Breyer estimated meant Brown had “three Cherokee ancestors at the time of George Washington’s father.”

Asked Roberts, who in the past has been critical of government classifications of race: “Is it one drop of blood that triggers all these extraordinary rights?”

The court also heard from lawyer Paul D. Clement, representing a guardian appointed by the South Carolina courts who thinks the Capobiancos should get Veronica. He said the case should be returned to a South Carolina court with instructions to place more emphasis on the best outcome for the child.

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